No Free Topology: Margin, Slope, and Architectural Cost in Deep Classifiers

Continuous separability alone does not reveal the architectural cost of reliable classification. We study this cost for scalar piecewise-linear classifiers: which decision distinctions must be preserved, and what resources does their realization require? A confidence-weighted margin deficit links recovered sign alternations to a necessary slope budget. With independent uniform threshold noise after the final score, the deficit of the clipped score equals twice the excess classification risk. For dyadic targets with $K$ boundaries, we obtain the exact cost $a{\log }_{2}K+b\lceil {\log }_2(1/(2r))\rceil$ at excess risk $0<r<1/8$ in a scalar catalog whose two-piece and affine modules cost $a\ge b>0$ and each have Lipschitz constant at most two. For irregular regions with unequal confidence, joint piece-slope bounds screen architecture budgets before optimization. Across 15 selection tasks, screening preserves every minimum-cost catalog choice while avoiding $38.0\%$ of dynamic-programming work. A complementary study uses finite diagnostics to choose repairs for information loss, metric compression, or insufficient resolution. On 270 clean synthetic tasks, it selects the same successful heads as cost-ordered search with half as many repair fits; label corruption and combined failures expose its limits. Together, these results motivate metric-topology factorization: preserve task-relevant distinctions, then realize their separation economically. The guarantees depend on the resource catalog and downstream noise model; increasing gain does not improve normalized input margins.